Quasifibrations of Graphs to Find Symmetries in Biological Networks

TL;DR

This paper introduces quasifibrations, a new theoretical framework and algorithmic approach to identify approximate symmetries and synchronization patterns in incomplete biological networks, overcoming limitations of traditional graph fibrations.

Contribution

It develops the theory of quasifibrations and provides an algorithm to detect near-symmetries in biological networks with missing or noisy data.

Findings

Algorithm successfully identifies quasifibrations in real connectome data.

Outperforms existing methods in repairing missing links.

Reconstructs incomplete networks with underlying symmetries.

Abstract

A fibration of graphs is an homomorphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. Recently, it has been shown that graph fibrations are useful tools to uncover symmetries and synchronization patterns in biological networks ranging from gene, protein,and metabolic networks to the brain. However, the inherent incompleteness and disordered nature of biological data precludes the application of the definition of fibration as it is; as a consequence, also the currently known algorithms to identify fibrations fail in these domains. In this paper, we introduce and develop systematically the theory of quasifibrations which attempts to capture more realistic patterns of almost-synchronization of units in biological networks. We provide an algorithmic solution to the problem of finding quasifibrations in networks where the existence of missing links and variability across samples preclude the identification of perfect symmetries in the connectivity structure. We test the algorithm against other strategies to repair missing links in incomplete networks using real connectome data and synthetic networks. Quasifibrations can be applied to reconstruct any incomplete network structure characterized by underlying symmetries and almost synchronized clusters.